Introducing General Relativity

An accessible and engaging introduction to general relativity for undergraduates

In Introducing General Relativity, the authors deliver a structured introduction to the core concepts and applications of General Relativity. The book leads readers from the basic ideas of relativity—including the Equivalence Principle and curved space-time—to more advanced topics, like Solar System tests and gravitational wave detection.

Each chapter contains practice problems designed to engage undergraduate students of mechanics, electrodynamics, and special relativity. A wide range of classical and modern topics are covered in detail, from exploring observational successes and astrophysical implications to explaining many popular principles, like space-time, redshift, black holes, gravitational waves and cosmology. Advanced topic sections introduce the reader to more detailed mathematical approaches and complex ideas, and prepare them for the exploration of more specialized and sophisticated texts.

Introducing General Relativity also offers:

• Structured outlines to the concepts of General Relativity and a wide variety of its applications
• Comprehensive explorations of foundational ideas in General Relativity, including space-time curvature and tensor calculus
• Practical discussions of classical and modern topics in relativity, from space-time to redshift, gravity, black holes, and gravitational waves
• Optional, in-depth sections covering the mathematical approaches to more advanced ideas

Perfect for undergraduate physics students who have studied mechanics, dynamics, and Special Relativity, Introducing General Relativity is an essential resource for those seeking an intermediate level discussion of General Relativity placed between the more qualitative books and graduate-level textbooks.

Mark Hindmarsh is Professor of Theoretical Physics with joint appointments at the University of Sussex, UK and the University of Helsinki, Finland. His research is focused on the physics of the Big Bang, and he is a member of the LISA consortium with particular expertise in the cosmological production of gravitational waves. He has taught at all levels of the undergraduate and postgraduate curriculum.

Andrew Liddle is a Principal Researcher at the University of Lisbon in Portugal, with joint affiliations at the University of Edinburgh, UK, and the Perimeter Institute for Theoretical Physics, Waterloo, Canada. He researches the properties of our Universe and how these relate to fundamental physical laws, especially through understanding astronomical observations. He is involved in several international projects, including the Planck Satellite and the Dark Energy Survey.

Preface ix

Constants and Symbols x

1 Introducing General Relativity 1

2 A Special Relativity Reminder 3

2.1 The need for Special Relativity 4

2.2 The Lorentz transformation 6

2.3 Time dilation 8

2.4 Lorentz–Fitzgerald contraction 9

2.5 Addition of velocities 11

2.6 Simultaneity, colocality, and causality 12

2.7 Space–time diagrams 13

3 Tensors in Special Relativity 17

3.1 Coordinates 18

3.2 4-vectors 20

3.3 4-velocity, 4-momentum, and 4-acceleration 24

3.4 4-divergence and the wave operator 26

3.5 Tensors 28

3.6 Tensors in action: the Lorentz force 30

4 Towards General Relativity 37

4.1 Newtonian gravity 37

4.2 Special Relativity and gravity 39

4.3 Motivations for a General Theory of Relativity 41

4.3.1 Mach’s Principle 42

4.3.2 Einstein’s Equivalence Principle 42

4.4 Implications of the Equivalence Principle 44

4.4.1 Gravitational redshift 45

4.4.2 Gravitational time dilation 46

4.5 Principles of the General Theory of Relativity 47

4.6 Towards curved space–time 49

4.7 Curved space in two dimensions 50

5 Tensors and Curved Space–Time 57

5.1 General coordinate transformations 57

5.2 Tensor equations and the laws of physics 59

5.3 Partial differentiation of tensors 59

5.4 The covariant derivative and parallel transport 60

5.5 Christoffel symbols of a two-sphere 65

5.6 Parallel transport on a two-sphere 66

5.7 Curvature and the Riemann tensor 68

5.8 Riemann curvature of the two-sphere 71

5.9 More tensors describing curvature 72

5.10 Local inertial frames and local flatness 73

6 Describing Matter 79

6.1 The Correspondence Principle 79

6.2 The energy–momentum tensor 80

6.2.1 General properties 80

6.2.2 Conservation laws and 4-vector flux 81

6.2.3 Energy and momentum belong in a rank-2 tensor 83

6.2.4 Symmetry of the energy–momentum tensor 84

6.2.5 Energy–momentum of perfect fluids 84

6.2.6 The energy–momentum tensor in curved space–time 87

7 The Einstein Equation 91

7.1 The form of the Einstein equation 91

7.2 Properties of the Einstein equation 93

7.3 The Newtonian limit 93

7.4 The cosmological constant 95

7.5 The vacuum Einstein equation 96

8 The Schwarzschild Space–time 99

8.1 Christoffel symbols 100

8.2 Riemann tensor 101

8.3 Ricci tensor 102

8.4 The Schwarzschild solution 103

8.5 The Jebsen–Birkhoff theorem 104

9 Geodesics and Orbits 109

9.1 Geodesics 109

9.2 Non-relativistic limit of geodesic motion 112

9.3 Geodesic deviation 113

9.4 Newtonian theory of orbits 115

9.5 Orbits in the Schwarzschild space–time 117

9.5.1 Massive particles 117

9.5.2 Photon orbits 120

10 Tests of General Relativity 123

10.1 Precession of Mercury’s perihelion 123

10.2 Gravitational light bending 125

10.3 Radar echo delays 127

10.4 Gravitational redshift 129

10.5 Binary pulsar PSR 1913+16 131

10.6 Direct detection of gravitational waves 135

11 Black Holes 139

11.1 The Schwarzschild radius 139

11.2 Singularities 140

11.3 Radial rays in the Schwarzschild space–time 141

11.4 Schwarzschild coordinate systems 143

11.5 The black hole space–time 145

11.6 Special orbits around black holes 147

11.7 Black holes in physics and in astrophysics 148

12 Cosmology 155

12.1 Constant-curvature spaces 156

12.2 The metric of the Universe 158

12.3 The matter content of the Universe 158

12.4 The Einstein equations 159

13 Cosmological Models 165

13.1 Simple solutions: matter and radiation 165

13.2 Light travel, distances, and horizons 169

13.2.1 Light travel in the cosmological metric 169

13.2.2 Cosmological redshift 170

13.2.3 The expansion rate 171

13.2.4 The age of the Universe 172

13.2.5 The distance–redshift relation and Hubble’s law 172

13.2.6 Cosmic horizons 173

13.2.7 The luminosity and angular-diameter distances 174

13.3 Ingredients for a realistic cosmological model 175

13.4 Accelerating cosmologies 180

14 General Relativity: The Next 100 Years 183

14.1 Developing General Relativity 183

14.2 Beyond General Relativity 184

14.3 Into the future 187

Advanced Topic A1 Geodesics in the Schwarzschild Space–Time 191

A1.1 Geodesics and conservation laws 191

A1.2 Schwarzschild geodesics for massive particles 192

A1.3 Schwarzschild geodesics for massless particles 194

Advanced Topic A2 The Solar System Tests in Detail 197

A2.1 Newtonian orbits in detail 197

A2.2 Perihelion shift in General Relativity 201

A2.3 Light deflection 204

A2.4 Time delay 205

Advanced Topic A3 Weak Gravitational Fields and Gravitational Waves 209

A3.1 Nearly-flat space–times 209

A3.2 Gravitational waves 211

A3.3 Sources of gravitational waves 214

Advanced Topic A4 Gravitational Wave Sources and Detection 219

A4.1 Gravitational waves from compact binaries 220

A4.2 The energy in gravitational waves 223

A4.3 Binary inspiral 224

A4.4 Detecting gravitational waves 227

A4.4.1 Laser interferometers 227

A4.4.2 Pulsar timing 230

A4.4.3 Interferometers in space 231

Bibliography 233

Answers to Selected Problems 237

Index 263

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